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Integrate exponentials of two bases over a power of x to the exponential integral - #1963
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Rafael-SOWNet merged 1 commit intoOct 20, 2026
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…ial integral a^x b^x/x was declined. The rules read one exponential, and a^x b^x is two of different bases that no spelling of the integrand merges: (a b)^x is e^(x ln(a b)), and ln(a b) is not ln(a) + ln(b) for every a and b. Beside a negative power of x, the product of exponentials with no quadratic left in the exponent is now integrated term by term: each power of x against e^(b x), by parts down to Ei(b x). A rate that cancels, b - b from e^(b x) e^(-b x), is decided by its value, and leaves a power of x alone. Rubi's 2.3 a^x b^x/x to a^x b^x/x^3. Part of #233. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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a^x b^x/xwas declined. The rules read one exponential, and no spelling of the integrand mergesa^x b^xinto one:(a b)^xise^(x ln(a b)), andln(a b)is notln(a) + ln(b)for everyaandb.Beside a negative power of x, a product of exponentials whose exponent has no quadratic left is now integrated term by term: each power of x against
e^(b x), by parts down toEi(b x). A rate that cancels,b - bfrome^(b x) e^(-b x), is decided by its value and leaves a power of x alone. A first version took that rate as written and divided by it, which gave two wrong answers. Those cases are in the test now.Measured on
8a1ef454, branch against master:Re-run alone on both builds, Rubi's 2.3:637–639 (
a^x b^x/xto/x^3) go from declined to answered in under 0.3 s. The other three moves time out on both builds. No run had a wrong answer.ExponentialsOfTwoBasesOverAPowerOfXIntegralTestdifferentiates the answers back.BREAKING-CHANGES.mdhas the entry, with the 2.5.0 column measured on v2.5.0.Part of #233.
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